Block-symmetric polynomials correlate with parity better than symmetric

Abstract

We show that degree-d block-symmetric polynomials in n variables modulo any odd p correlate with parity exponentially better than degree-d symmetric polynomials, if ${n \ge cd^2 {\rm log} d}$ and ${d \in [0.995 \cdot p^t - 1,p^t)}$ for some ${t \ge 1}$ and some ${c > 0}$ that depends only on p. For these infinitely many degrees, our result solves an open problem raised by a number of researchers including Alon & Beigel (IEEE conference on computational complexity (CCC), pp 184–187, 2001). The only previous case for which this was known was d = 2 and p = 3 (Green in J Comput Syst Sci 69(1):28–44, 2004).

The result is obtained through the development of a theory we call spectral analysis of symmetric correlation, which originated in works of Cai et al. (Math Syst Theory 29(3):245–258, 1996) and Green (Theory Comput Syst 32(4):453–466, 1999). In particular, our result follows from a detailed analysis of the correlation of symmetric polynomials, which is determined up to an exponentially small relative error when ${d = p^t-1}$.